View source for Sampling (signal processing)

{{Short description|Measuring a continuous signal at a series of instants}}
{{Hatnote|For the converter that does the measuring, see [[Analog-to-digital converter]].}}
'''Sampling''' is the measurement of a continuous signal at a series of instants, usually equally spaced. The number of measurements each second is the sampling rate, ''f''<sub>s</sub>, and the time between them is 1/''f''<sub>s</sub>. A nanopore instrument samples the [[ionic current]] through a pore: at 200 kHz, the rate of the amplifier on the company's [[live molecular stream]], a measurement is made every 5 µs, and a molecule that blocks the pore for 50 µs is seen in ten of them.<ref name="faq">{{cite web |url=https://molecularreality.com/FAQ |title=Keltar's Field Guide to Molecular Reality |publisher=Molecular Reality Corporation |date=August 2026}}</ref> The question that sampling theory answers is when those ten numbers are enough to say what the current did in between.

== The theorem ==
A signal that contains no frequencies above ''B'' is fixed completely by samples taken more than 2''B'' times a second. Half the sampling rate, ''f''<sub>s</sub>/2, is the Nyquist frequency. The result has several fathers. E. T. Whittaker studied the interpolation series in 1915; Harry Nyquist showed in 1928 that a telegraph channel of bandwidth ''B'' can carry no more than 2''B'' independent pulses a second;<ref name="nyquist">{{cite journal |last1=Nyquist |first1=H. |title=Certain topics in telegraph transmission theory |journal=Transactions of the American Institute of Electrical Engineers |volume=47 |issue=2 |pages=617-644 |year=1928 |doi=10.1109/T-AIEE.1928.5055024}}</ref> Vladimir Kotelnikov proved the theorem in 1933; and Claude Shannon stated it as a tool for communication engineers in 1949, in the form usually taught.<ref name="shannon">{{cite journal |last1=Shannon |first1=C. E. |title=Communication in the presence of noise |journal=Proceedings of the IRE |volume=37 |issue=1 |pages=10-21 |year=1949 |doi=10.1109/JRPROC.1949.232969}}</ref> Shannon's statement includes the recipe for getting the signal back: place a curve of the form sin(π''x'')/(π''x'') at each sample, scale it by the sample, and add them up.

== Aliasing ==
A frequency above the Nyquist frequency is not lost. It appears in the samples at a lower, false frequency, because a fast sine wave and a slow one can pass through the same row of points. A component at ''f'' shows up at |''f'' − ''k'' ''f''<sub>s</sub>|, with ''k'' the whole number that brings it into the band from zero to ''f''<sub>s</sub>/2. At 200 kS/s a component at 150 kHz is recorded as one at 50 kHz, and nothing in the record tells the two apart. The [[Analog-to-digital converter]] article shows the case of 70 kHz sampled at 100 kS/s, which arrives at 30 kHz.

The cure is to remove what is too fast before the converter sees it, with an analog low-pass filter. Because a filter does not cut off sharply, the cutoff is set well below the Nyquist frequency. Papers on solid-state pores report filter cutoffs of 0.1 to 0.4 of the sampling rate, with a [[Bessel filter]] the usual choice because it does not distort the shape of a pulse.<ref name="kwok">{{cite journal |last1=Kwok |first1=H. |last2=Briggs |first2=K. |last3=Tabard-Cossa |first3=V. |title=Nanopore fabrication by controlled dielectric breakdown |journal=PLOS ONE |volume=9 |issue=3 |pages=e92880 |year=2014 |doi=10.1371/journal.pone.0092880}}</ref><ref name="smeets">{{cite journal |last1=Smeets |first1=R. M. M. |last2=Keyser |first2=U. F. |last3=Dekker |first3=N. H. |last4=Dekker |first4=C. |title=Noise in solid-state nanopores |journal=Proceedings of the National Academy of Sciences |volume=105 |issue=2 |pages=417-421 |year=2008 |doi=10.1073/pnas.0705349105}}</ref> A step in current, which is what an event is, contains frequencies at every height, so no sampling rate is high enough to record a pore's events without a filter. The filter decides what is kept, and the sampling rate has to be chosen to suit it.

== Choosing a rate ==
[[File:Event_sampled_at_two_rates.svg|thumb|right|upright=1.5|alt=Two panels showing the same filtered 50 microsecond dip in current, one sampled every 5 microseconds with ten points inside the dip and one every 10 microseconds with five|A blockade of 50 µs, filtered with a single pole at 70 kHz and sampled every 5 µs (top) and every 10 µs (bottom). Calculated; no noise is drawn.]]
Three things pull in different directions. Faster sampling resolves shorter events and lets a gentler filter pass them. It also produces more data: a 16-bit record at 100 kS/s is 200 kB a second, 720 MB an hour. And a wider filter admits more noise (see [[Signal-to-noise ratio]]). Commercial sequencers sample slowly, because a motor protein holds the strand: a 2018 chemistry advanced the strand at 450 bases a second and was sampled at 4 kHz, about nine samples to a base.<ref name="rang">{{cite journal |last1=Rang |first1=F. J. |last2=Kloosterman |first2=W. P. |last3=de Ridder |first3=J. |title=From squiggle to basepair: computational approaches for improving nanopore sequencing read accuracy |journal=Genome Biology |volume=19 |pages=90 |year=2018 |doi=10.1186/s13059-018-1462-9}}</ref> A solid-state pore passes DNA thousands of times faster and has no such luxury.

Sampling faster than the filter requires has one use: the extra samples can be averaged. Averaging blocks of 100 samples turns a 100 kS/s record into a 1 kS/s one with white noise lower by a factor of 10, which is how the web display of the [[MR1 Nanopore Reader]] is fed.<ref name="records">{{cite web |title=Company records |publisher=Molecular Reality Corporation |date=October 2026}}</ref> Oversampling in this way spreads the noise of the converter's rounding over a wider band, and each fourfold increase in rate buys about one bit of resolution after averaging.

== At Molecular Reality ==
The company's instruments sample at 200 kHz on the live stream, at 100 kS/s in the MR1 Nanopore Reader and, in the first batch of amplifiers for the [[demonpore 64]], at "up to 125,000 samples of current per second".<ref name="records" /> That last rate carries a signal only to 62.5 kHz. Elsewhere the figure 125 kHz is given as the bandwidth of the amplifier, which would call for more than 250,000 samples a second, a point worked through in [[Teabiscuits]]. The README of the MR1 repository labels the link between the amplifier board and its converter "SPI Bus (100 kHz)", the sample rate and not the clock of the bus, which for 16-bit samples must run at 1.6 MHz or more.<ref name="github">{{cite web |url=https://github.com/EpicQuestBio/MR1 |title=MR1 Nanopore Reader |website=GitHub |date=October 16, 2026}}</ref>

The company's critique of its own plans for machine learning compares the rates of two instruments it proposes to combine: a camera at 30 frames a second sees the flow cell once every 33,000 µs, and the amplifier samples it every 5 µs, a ratio of about 6,600 to 1 ([[The Nanopore World Model]]).<ref name="records" />

== See also ==
* [[Bandwidth (signal processing)]]
* [[Analog front end]]
* [[EDR4]]

== References ==
{{Reflist}}

{{Nanopores}}

[[Category:Signal processing]]
[[Category:Nanopore sensing]]